# BCT

##### Box-Cox t distribution for fitting a GAMLSS

The function `BCT()`

defines the Box-Cox t distribution, a four parameter distribution,
for a `gamlss.family`

object to be used in GAMLSS fitting using the function `gamlss()`

. The functions `dBCT`

,
`pBCT`

, `qBCT`

and `rBCT`

define the density, distribution function, quantile function and random
generation for the Box-Cox t distribution.
[The function `BCTuntr()`

is the original version of the function suitable only for the untruncated BCT distribution].
See Rigby and Stasinopoulos (2003) for details.
The function `BCT`

is identical to `BCT`

but with log link for mu.

- Keywords
- distribution, regression

##### Usage

```
BCT(mu.link = "identity", sigma.link = "log", nu.link = "identity",
tau.link = "log")
BCTo(mu.link = "log", sigma.link = "log", nu.link = "identity",
tau.link = "log")
BCTuntr(mu.link = "identity", sigma.link = "log", nu.link = "identity",
tau.link = "log")
dBCT(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCT(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
qBCT(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
rBCT(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)
dBCTo(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCTo(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
qBCTo(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
rBCTo(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)
```

##### Arguments

- mu.link
Defines the

`mu.link`

, with "identity" link as the default for the`mu`

parameter. Other links are "inverse", "log" and "own"- sigma.link
Defines the

`sigma.link`

, with "log" link as the default for the`sigma`

parameter. Other links are "inverse","identity", "own"- nu.link
Defines the

`nu.link`

, with "identity" link as the default for the`nu`

parameter. Other links are "inverse", "log", "own"- tau.link
Defines the

`tau.link`

, with "log" link as the default for the`tau`

parameter. Other links are "inverse", "identity" and "own"- x,q
vector of quantiles

- mu
vector of location parameter values

- sigma
vector of scale parameter values

- nu
vector of

`nu`

parameter values- tau
vector of

`tau`

parameter values- log, log.p
logical; if TRUE, probabilities p are given as log(p).

- lower.tail
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]

- p
vector of probabilities.

- n
number of observations. If

`length(n) > 1`

, the length is taken to be the number required

##### Details

The probability density function of the untruncated Box-Cox t distribution, `BCTuntr`

, is given by
$$f(y|\mu,\sigma,\nu,\tau)=\frac{y^{\nu-1}}{\mu^{\nu}\sigma} \frac{\Gamma[(\tau+1)/2]}{\Gamma(1/2) \Gamma(\tau/2) \tau^{0.5}} [1+(1/\tau)z^2]^{-(\tau+1)/2}$$
where if \(\nu \neq 0\) then \(z=[(y/\mu)^{\nu}-1]/(\nu \sigma)\) else \(z=\log(y/\mu)/\sigma\),
for \(y>0\), \(\mu>0\), \(\sigma>0\), \(\nu=(-\infty,+\infty)\) and \(\tau>0\).

The Box-Cox *t* distribution, `BCT`

, adjusts the above density \(f(y|\mu,\sigma,\nu,\tau)\) for the
truncation resulting from the condition \(y>0\). See Rigby and Stasinopoulos (2003) for details.

##### Value

`BCT()`

returns a `gamlss.family`

object which can be used to fit a Box Cox-t distribution in the `gamlss()`

function.
`dBCT()`

gives the density, `pBCT()`

gives the distribution
function, `qBCT()`

gives the quantile function, and `rBCT()`

generates random deviates.

##### Note

\(\mu\) is the median of the distribution, \(\sigma(\frac{\tau}{\tau-2})^{0.5}\)
is approximate the coefficient of variation (for small \(\sigma\) and moderate `nu>0`

and moderate or large \(\tau\)),
\(\nu\) controls the skewness and \(\tau\) the kurtosis of the distribution

##### Warning

The use `BCTuntr`

distribution may be unsuitable for some combinations of the parameters (mainly for large \(\sigma\))
where the integrating constant is less than 0.99. A warning will be given if this is the case.

The `BCT`

distribution is suitable for all combinations of the parameters within their ranges
[i.e. \(\mu>0,\sigma>0, \nu=(-\infty,\infty) {\rm and} \tau>0\) ]

##### References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion),
*Appl. Statist.*, **54**, part 3, pp 507-554.

Rigby, R.A. Stasinopoulos, D.M. (2006). Using the Box-Cox *t* distribution in GAMLSS to mode skewnees and and kurtosis.
to appear in *Statistical Modelling*.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019)
*Distributions for modeling location, scale, and shape: Using GAMLSS in R*, Chapman and Hall/CRC. An older version can be found in http://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R.
*Journal of Statistical Software*, Vol. **23**, Issue 7, Dec 2007, http://www.jstatsoft.org/v23/i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017)
*Flexible Regression and Smoothing: Using GAMLSS in R*, Chapman and Hall/CRC.

##### See Also

##### Examples

```
# NOT RUN {
BCT() # gives information about the default links for the Box Cox t distribution
# library(gamlss)
#data(abdom)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=BCT, data=abdom) #
#plot(h)
plot(function(x)dBCT(x, mu=5,sigma=.5,nu=1, tau=2), 0.0, 20,
main = "The BCT density mu=5,sigma=.5,nu=1, tau=2")
plot(function(x) pBCT(x, mu=5,sigma=.5,nu=1, tau=2), 0.0, 20,
main = "The BCT cdf mu=5, sigma=.5, nu=1, tau=2")
# }
```

*Documentation reproduced from package gamlss.dist, version 5.1-7, License: GPL-2 | GPL-3*